Examples 10.1:

(i) The closed unit interval $[0, 1]$ has the fixed point property (exercise 1, lecture 3).

(ii) A non-trivial topological group does not have the fixed point property.

(iii) The space $\mathbb{R}P^{2n}$ has the fixed point property but we are not yet ready to prove this.

(iv) The open unit disc $U = \{z\in \mathbb{C}/\vert z\vert < 1\}$ does not have the fixed point property. For if $a$ is a non-zero complex number with $\vert a\vert < 1$ then the map $f:U\longrightarrow U$ given by

$$f(z) = \frac{z-a}{1-\overline{a}z}$$

has no fixed points in $U$. The reader must first check that $f$ maps the open unit disc to itself and examine if it has any fixed points.